Four large technology companies should not be allowed to limit evidence about Apple Inc co-founder Steve Jobs at an upcoming trial over no-hire agreements in Silicon Valley, according to a court document filed late on Thursday by employees suing the firms.

Tech workers brought a class action lawsuit against Apple, Google Inc, Intel Inc and Adobe Systems Inc in 2011, alleging they conspired to avoid competing for each other’s employees in order to avert a salary war. The trial is scheduled to begin at the end of May on behalf of roughly 64,000 workers in the class, and plaintiffs say damages could top $3 billion.

The case, which is closely watched in Silicon Valley, is largely built on emails among top executives, including Apple’s late chief executive Jobs and former Google CEO Eric Schmidt.

In one instance, after a Google recruiter solicited an Apple employee, Schmidt told Jobs that the recruiter would be fired, court documents show. Jobs then forwarded Schmidt’s note to a top Apple human resources executive with a smiley face appended.

Earlier this week, the four companies asked US district judge Lucy Koh to prevent plaintiffs from unfairly portraying Jobs as a “bully” at trial. The companies said they did not seek to bar Jobs’ communications about the no-hire agreements, but rather evidence gleaned from sources like Walter Isaacson’s bestselling biography about Jobs.

However, in the filing on Thursday, the employees said such material had been used in separate antitrust litigation involving Apple over e-books.

“That the jury might draw conclusions about Jobs’ character based on evidence showing the manner in which he pursued the conspiracy at the heart of this case is not grounds to exclude such evidence,” they wrote.

A Google spokesman declined to comment. Representatives for Apple, Intel and Adobe could not immediately be reached for comment, nor could an attorney for the plaintiffs.

The four companies agreed to settle a US Department of Justice probe in 2010 that barred them from entering into such no-hire agreements in the future. They have since been fighting the civil antitrust class action, arguing that the plaintiffs cannot successfully prove an overarching conspiracy to impact wages.

In addition to evidence about Jobs’ character, the tech companies also asked Koh to prohibit discussion of the government probe at trial. However, on Thursday the plaintiffs argued that the jury should know the reason the companies eliminated their no-hire agreements.

Additionally, the plaintiffs seek to introduce evidence about the personal wealth of executives like Google co-founder Sergey Brin and how it could be enhanced by holding down workers’ salaries and boosting margins, according to the filing.